We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global mild solutions in bounded domains of R N , N = 2, 3, for small initial data in L p -spaces.− Attraction-repulsion chemotaxis model without fluid and logistic source System (1.1) without fluid and logistic source (u = 0, ς = µ = κ 2 = 0, κ 1 = 1) has been analyzed by several authors, see for instance [18,19,20,28,30]. This model corresponds to a direct generalization of the classical Keller-Segel chemotaxis system; however, the analysis of the large time behavior of solutions is difficult due to the lack of a Lyapunov functional. For one-dimensional bounded domains, proved the existence of global classical solutions based on Amann's theory and the method of energy estimates. For a more recent result in one dimension, see [19]. For twodimensional bounded domains, Jin and Wang [20] considered a parabolic-parabolic-elliptic case where
In this paper we develop a numerical scheme for approximating a $d$-dimensional chemotaxis-Navier-Stokes system, $d=2,3$, modeling cellular swimming in incompressible fluids. This model describes the chemotaxis-fluid interaction in cases where the chemical signal is consumed with a rate proportional to the amount of organisms. We construct numerical approximations based on the Finite Element method and analyze optimal error estimates and convergence towards regular solutions. In order to construct the numerical scheme, we use a splitting technique to deal with the chemo-attraction term in the cell-density equation, leading to introduce a new variable given by the gradient of the chemical concentration. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed and mass-conservative. We obtain uniform estimates and analyze the convergence of the scheme. Finally, we present some numerical simulations to verify the good behavior of our scheme, as well as to check numerically the optimal error estimates proved in our theoretical analysis.
In this paper we develop a numerical scheme for approximating a d-dimensional chemotaxis-Navier-Stokes system, d = 2, 3, modeling cellular swimming in incompressible fluids. This model describes the chemotaxis-fluid interaction in cases where the chemical signal is consumed with a rate proportional to the amount of organisms. We construct numerical approximations based on the Finite Element method and analyze some error estimates and convergence towards weak solutions. In order to construct the numerical scheme, we use a splitting technique to deal with the chemo-attraction term in the cell-density equation, leading to introduce a new variable given by the gradient of the chemical concentration. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed and it is mass-conservative. We obtain uniform estimates and analyze the convergence of the scheme. Finally, we present some numerical simulations to verify the good behavior of our scheme, as well as to check numerically the error estimates proved in our theoretical analysis.
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